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Fractions are generally used to define any whole number into equal parts. While writing a fraction, there are two numbers involved. The number at the top is called the numerator, while that at the bottom is called the denominator. There are three types of fractions:

  • Proper Fractions: In proper fractions, the denominator is greater than the numerator. For example: \(\frac{3}{7}, \frac{1}{3}\)
  • Improper Fractions: As the name suggests, these fractions are “top-heavy” where the numerator is greater than the denominator. For example: \(\frac{7}{3}, \frac{5}{2}\)
  • Mixed Fractions: Mixed fractions are another type of improper fraction where there is a whole number as well as a fractional part. For example: \(1 \frac{1}{3}, 2 \frac{3}{7}\)

Multiplication of 2 Fractions

Let’s take the example of two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\).

\(\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\)

So, multiplication is pretty simple. Just multiply the two numerators and write the result over the multiplication result of the two denominators.

Note: Before directly multiplying fractions, you can convert each one of them into their simplest forms. Also, you can even cross out equal factors to make each number as small as possible.

Division of 2 Fractions

Let’s take the example of two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\).

\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c}\)

Division too, is pretty simple. Just write the first fraction as it is, and then flip the second fraction. Once flipped, change the sign from division to multiplication. Now proceed as normal.

Finally, you may have a fraction that can be reduced to a simpler form. It’s always best to reduce to the simplest form when possible.

Exercises for Multiplying and Dividing Fractions

  1. \(\frac{2}{3} \times \frac{17}{15} = \)
  2. \(\frac{3}{2} \times \frac{10}{16} = \)
  3. \(\frac{4}{6} \times \frac{18}{17} = \)
  4. \(\frac{7}{10} \times \frac{7}{8} = \)
  5. \(\frac{7}{3} \times \frac{1}{5} = \)
  6. \(\frac{9}{7} \div \frac{6}{5} = \)
  7. \(\frac{10}{5} \div \frac{8}{3} = \)
  8. \(\frac{6}{3} \div \frac{14}{18} = \)
  9. \(\frac{8}{10} \div \frac{16}{17} = \)
  10. \(\frac{8}{7} \div \frac{15}{20} = \)
  11. \(\frac{1}{6} \div \frac{3}{7} = \)
  12. \(\frac{1}{6} \div \frac{3}{6} = \)

Answers

  1. \(\frac{2}{3} \times \frac{17}{15} = \frac{2 \times 17}{3 \times 15} = \frac{34}{45}\)

    Solution:

    Multiply the top numbers, and then multiply the bottom numbers.

    \(\frac{2}{3} \times \frac{17}{15} = \frac{2 \times 17}{3 \times 15} = \frac{34}{45}\)

  2. \(\frac{3}{8} \times \frac{10}{16} = \frac{3 \times 10}{8 \times 16} = \frac{30}{128}\)

    Solution:

    Step 1: Multiply the top numbers, and then multiply the bottom numbers.

    \(\frac{3}{8} \times \frac{10}{16} = \frac{3 \times 10}{8 \times 16} = \frac{30}{128}\)

    Step 2: Simplify your answer.

    \(\frac{30}{128} = \frac{30 \div 2}{128 \div 2} = \frac{15}{64}\)

    \(\text{GCF}(30, 128) = 2\)

  3. \(\frac{4}{6} \times \frac{18}{17} = \frac{4 \times 18}{6 \times 17} = \frac{72}{102}\)

    Solution:

    Step 1: Multiply the top numbers, and then multiply the bottom numbers.

    \(\frac{4}{6} \times \frac{18}{17} = \frac{4 \times 18}{6 \times 17} = \frac{72}{102}\)

    Step 2: Simplify your answer.

    \(\frac{72}{102} = \frac{72 \div 6}{102 \div 6} = \frac{12}{17}\)

    \(\text{GCF}(72, 102) = 6\)

  4. \(\frac{7}{10} \times \frac{7}{8} = \frac{7 \times 7}{10 \times 8} = \frac{49}{80}\)

    Solution:

    Multiply the top numbers, and then multiply the bottom numbers.

    \(\frac{7}{10} \times \frac{7}{8} = \frac{7 \times 7}{10 \times 8} = \frac{49}{80}\)

  5. \(\frac{7}{3} \times \frac{1}{5} = \frac{7 \times 1}{3 \times 5} = \frac{7}{15}\)

    Solution:

    Multiply the top numbers, and then multiply the bottom numbers.

    \(\frac{7}{3} \times \frac{1}{5} = \frac{7 \times 1}{3 \times 5} = \frac{7}{15}\)

  6. \(\frac{9}{7} \times \frac{6}{5} = \frac{9 \times 6}{7 \times 5} = \frac{54}{35}\)

    Solution:

    Multiply the top numbers, and then multiply the bottom numbers.

    \(\frac{9}{7} \times \frac{6}{5} = \frac{9 \times 6}{7 \times 5} = \frac{54}{35}\)

  7. \(\frac{10}{3} \div \frac{5}{8} = \frac{10 \times 8}{3 \times 5} = \frac{80}{15} = \frac{16}{3}\)

    Solution:

    Step 1: Keep the first fraction, change the division sign to multiplication, and flip the numerator and denominator of the second fraction. Then multiply them.

    \(\frac{10}{3} \div \frac{5}{8} = \frac{10 \times 8}{3 \times 5} = \frac{80}{15}\)

    Step 2: Simplify your answer.

    \(\frac{80}{15} = \frac{16}{3}\)

  8. \(\frac{6}{14} \div \frac{18}{42} = \frac{6 \times 42}{14 \times 18} = \frac{252}{252} = \frac{18}{7}\)

    Solution:

    Step 1: Keep the first fraction, change the division sign to multiplication, and flip the numerator and denominator of the second fraction. Then multiply them.

    \(\frac{6}{14} \div \frac{18}{42} = \frac{6 \times 42}{14 \times 18} = \frac{252}{252}\)

    Step 2: Simplify your answer.

    \(\frac{252}{252} = \frac{18}{7}\)

  9. \(\frac{6}{17} \div \frac{16}{10} = \frac{6 \times 10}{17 \times 16} = \frac{60}{272} = \frac{15}{68}\)

    Solution:

    Step 1: Keep the first fraction, change the division sign to multiplication, and flip the numerator and denominator of the second fraction. Then multiply them.

    \(\frac{6}{17} \div \frac{16}{10} = \frac{6 \times 10}{17 \times 16} = \frac{60}{272}\)

    Step 2: Simplify your answer.

    \(\frac{60}{272} = \frac{15}{68}\)

  10. \(\frac{8}{20} \div \frac{15}{7} = \frac{8 \times 7}{20 \times 15} = \frac{56}{300} = \frac{14}{75}\)

    Solution:

    Step 1: Keep the first fraction, change the division sign to multiplication, and flip the numerator and denominator of the second fraction. Then multiply them.

    \(\frac{8}{20} \div \frac{15}{7} = \frac{8 \times 7}{20 \times 15} = \frac{56}{300}\)

    Step 2: Simplify your answer.

    \(\frac{56}{300} = \frac{14}{75}\)

  11. \(\frac{2}{5} \div \frac{3}{2} = \frac{2 \times 2}{5 \times 3} = \frac{4}{15}\)

    Solution:

    Step 1: Keep the first fraction, change the division sign to multiplication, and flip the numerator and denominator of the second fraction. Then multiply them:

    \(\frac{2}{5} \div \frac{3}{2} = \frac{2 \times 2}{5 \times 3} = \frac{4}{15}\)

  12. \(\frac{1}{6} \div \frac{6}{3} = \frac{1 \times 3}{6 \times 6} = \frac{3}{36} = \frac{1}{12}\)

    Solution:

    Step 1: Keep the first fraction, change the division sign to multiplication, and flip the numerator and denominator of the second fraction. Then multiply them:

    \(\frac{1}{6} \div \frac{6}{3} = \frac{1 \times 3}{6 \times 6} = \frac{3}{36}\)

    Step 2: Simplify your answer:

    \(\frac{3}{36} = \frac{1}{12}\)